Theorem ttcwf2 36890

Description: If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcwf2	⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem ttcwf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)))
2	1	eldifad 3917	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ TC+ 𝐴)
3		ttctr2 36859	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
4	2, 3	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ TC+ 𝐴)
5		dfss2 3923	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
6	4, 5	sylib 220	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
7		inssdif0 4328	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On) ↔ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
8	7	bilanri 510	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On))
9	6, 8	eqsstrrd 3972	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ ∪ (𝑅1 “ On))
10		vex 3459	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
11	10	r1elss 9762	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On))
12	9, 11	sylibr 236	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ ∪ (𝑅1 “ On))
13	1	eldifbd 3918	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → ¬ 𝑥 ∈ ∪ (𝑅1 “ On))
14	12, 13	pm2.65da 826	. . . . . . . 8 ⊢ (𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) → ¬ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
15	14	nrex 3091	. . . . . . 7 ⊢ ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅
16	15	a1i 11	. . . . . 6 ⊢ (TC+ 𝐴 ∈ V → ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
17		difexg 5286	. . . . . . 7 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V)
18		zfreg 9542	. . . . . . 7 ⊢ (((TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
19	17, 18	sylan 589	. . . . . 6 ⊢ ((TC+ 𝐴 ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
20	16, 19	mtand 825	. . . . 5 ⊢ (TC+ 𝐴 ∈ V → ¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅)
21		nne 2962	. . . . 5 ⊢ (¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅ ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
22	20, 21	sylib 220	. . . 4 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
23		ssdif0 4320	. . . 4 ⊢ (TC+ 𝐴 ⊆ ∪ (𝑅1 “ On) ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
24	22, 23	sylibr 236	. . 3 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))
25		eleq1 2851	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On)))
26		sseq1 3962	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ⊆ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
27	25, 26	bibi12d 347	. . . 4 ⊢ (𝑥 = TC+ 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) ↔ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))))
28	27, 11	vtoclg 3523	. . 3 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
29	24, 28	mpbird 259	. 2 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
30		elex 3476	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ V)
31	29, 30	impbii 211	1 ⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4866   “ cima 5651  Oncon0 6346  𝑅₁cr1 9718  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-reg 9538

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-r1 9720  df-ttc 36852

This theorem is referenced by:  ttcwf3  36891